Month: January 2014

Fearing our buses wouldn’t start due to cold, our district let them idle overnight. The first student question this morning: “How much did that cost?”

That’s kind of amazing.

There’s a local, personally relevant, real-world math problem somewhere in there for students to work on and learn from. But one of my theses with fake-world math is that relevance and the “real world” aren’t necessary or sufficient. They don’t guarantee interest and they don’t guarantee learning.

So tell me about an effective treatment of this situation in math class. (Draw on research on curiosity, abstraction, and the CCSS modeling framework if they’re helpful.) Also tell me about an ineffective treatment of this situation in math class.

BTW. “Curriculum Confab” will be a recurring feature around here, similar to our early “What Can You Do With This?” days only with more design and theory attached.

Today Desmos is releasing Function Carnival, an online math happytime we spent several months developing in collaboration with Christopher Danielson. Christopher and I drafted an announcement over at Desmos which summarizes some research on function misconceptions and details our efforts at addressing them. I hope you’ll read it but I don’t want to recap it here.

Instead, I’d like to be explicit about three claims we’re making about online math education with Function Carnival.

1. We can ask students to do lots more than fill in blanks and select from multiple choices.

Currently, students select from a very limited buffet line of experiences when they try to learn math online. They watch videos. They answer questions about what they watched in the videos. If the answer is a real number, they’re asked to fill in a blank. If the answer is less structured than a real number, we often turn to multiple choice items. If the answer is something even less structured, something like an argument or a conjecture … well … students don’t really do those kinds of things when they learn math online, do they?

With Function Carnival, we ask students to graph something they see, to draw a graph by clicking with their mouse or tapping with their finger.

We also ask students to make arguments about incorrect graphs.

I’d like to know another online math curriculum that assigns students the tasks of drawing graphs and arguing about them. I’m sure it exists. I’m sure it isn’t common.

2. We can give students more useful feedback than “right/wrong” with structured hints.

Currently, students submit an answer and they’re told if it’s right or wrong. If it’s wrong, they’re given an algorithmically generated hint (the computer recognizes you probably got your answer by multiplying by a fraction instead of by its reciprocal and suggests you check that) or they’re shown one step at a time of a worked example (“Here’s the first step for solving a proportion. Do you want another?”).

This is fine to a certain extent. The answers to many mathematical questions are either right or wrong and worked examples can be helpful. But a lot of math questions have many correct answers with many ways to find those answers and many better ways to help students with wrong answers than by showing them steps from a worked example.

For example, with Function Carnival, when students draw an incorrect graph, we don’t tell them they’re right or wrong, though that’d be pretty simple. Instead, we echo their graph back at them. We bring in a second cannon man that floats along with their graph and they watch the difference between their cannon man and the target cannon man. Echoing. (Or “recursive feedback” to use Okita and Schwartz’s term.)

When I taught with Function Carnival in two San Jose classrooms, the result was students who would iterate and refine their graphs and often experience useful realizations along the way that made future graphs easier to draw.

3. We can give teachers better feedback than columns filled with percentages and colors.

Our goal here isn’t to distill student learning into percentages and colors but to empower teachers with good data that help them remediate student misconceptions during class and orchestrate productive mathematical discussions at the end of class. So we take in all these student graphs and instead of calculating a best-fit score and allowing teachers to sort it, we built filters for common misconceptions. We can quickly show a teacher which students evoke those misconceptions about function graphs and then suggest conversation starters.

A bonus claim to play us out:

4. This stuff is really hard to do well.

Maybe capturing 50% the quality of our best brick-and-mortar classrooms at 25% the cost and offering it to 10,000% more people will win the day. Before we reach that point, though, let’s put together some existence proofs of online math activities that capture more quality, if also at greater cost. Let’s run hard and bury a shoulder in the mushy boundary of what we call online math education, then back up a few feet and explore the territory we just revealed. Function Carnival is our contribution today.

Provoking curiosity in our students about anything requires us to manage several tensions simultaneously. It requires keeping several lines tight – not slack – but not so tight they snap.

Read on for recommendations from some careful researchers.

Previously

Here is where the series stands: I’ve suggested that educators dramatically overvalue the real world as a motivator for students (one example) and that pinning down a definition of what is “real” to a child is no light assignment. I’ve suggested, instead, that the purpose of math class is to build a student’s capacity to puzzle and unpuzzle herself. And we shouldn’t limit the source of those puzzles. They can come from anywhere, including the world of pure mathematics. As an existence proof, I listed some abstract experiences that humans have enjoyed (everything from Sudoku to the Four Fours).

Currently

This is a bunch of question-begging, though, and my commenters have rightly called me out:

Okay, bud, how do you turn the “boring” bits of math into puzzles?

We struggle here. Across three conferences this fall, I’ve received different answers from respected educators about how we should handle the boring bits of math. (No one offers a definition of “boring,” by the way, but I take it to mean questions about mathematical abstractions – numbers, variables, etc. – instead of the material world.)

One educator suggested we “flip” those boring bits and send them home in a digital video. Another suggested we ask for sympathy, telling kids, “Hey, it can’t all be fun, okay? Just go with me here.” Another suggested we aim for empathy, that by accentuating our own enthusiasm for boring material our students might follow our lead. These answers all require you to believe there are mathematical concepts that are irredeemably boring, that there are aspects of the world about which we can’t possibly be curious. I don’t.

So I spent my holiday reading about curiosity, starting with some bread crumbs laid out by Annie Murphy Paul. These are some lessons I learned about teaching “the boring bits.”

Disclaimer

Interest and curiosity aren’t binary variables. They aren’t “on” or “off.” In his research on curiosity, Paul Silvia noted that “people differ in whether they find something interesting” and that “the same person will differ in interest over time” (2008, p. 58).

These could all fit in a tweet. In half a tweet. They’re light on disciplinary language, which keeps them comprehensible. In the final two tasks, students maintain a sense of personal control as they select individual starting points for each task.

Start an argument.

In 1981, Smith, Johnson & Johnson ran an experiment that resulted in one group of students skipping recess to learn new concepts and another group proceeding outside as usual, uninterested in learning those same concepts. The difference was controversy. The researchers engineered arguments between students in the first group, but not in the second.

Can you engineer arguments between students about the boring bits of mathematics?

Is zero even or odd?

Does multiplying numbers always make them bigger?

Can you create a system of equations that has no solution?

True or false: doubling the perimeter of a shape doubles its area.

At NCTM’s annual conference in Denver, Steve Leinwand said “the most important nine words of the Common Core State Standards are ‘construct viable arguments and critique the reasoning of others’.”

So not only can arguments stir your students’ curiosity but they’re an essential part of their math education. That’s winning twice.

Engineer a counterintuitive moment.

Hunt (1963, 1965) and Kagan (1972) popularized the “incongruity” account of curiosity. George Loewenstein summarizes: “People tend to be curious about events that are unexpected or that they cannot explain” (1994, p. 83). These events are difficult to engineer, of course, because they depend on the knowledge a student brings into your classroom. You have to know what your students expect in order to show them something they don’t expect.

Here are several existence proofs:

The Magic Octagon. The arrow isn’t where the student thought it would be. “Wait, what?”

These moments are everywhere, though it’s an ongoing effort to train my eyes to find them. You can find them when the world becomes unexpectedly orderly or unexpectedly disorderly. When we all choose numbers that add to five and graph them, we get an unexpectedly orderly line. When we try to apply a proportional model to footage of a water tank emptying (“It took five minutes to empty halfway so it’ll take ten minutes to empty all the way.”) the world becomes unexpectedly disorderly. For younger students, the fact that 2 + 3 is the same as 3 + 2 may be a moment of curiosity and counterintuition. You know you landed the moment because the expression flashes across your student’s face: “Wait. What?”

You also create a counterintuitive moment when you …

Break their old tools.

Students bring functional tools into your classroom. They may know how to count sums on their fingers. They may know how to calculate the slope of a line by counting unit-squares and dividing the vertical squares by the horizontal squares. They may know how to write down and recall small numbers.

You create counterintuition when you take those old, functional tools and assign them to a task which initially seems appropriate but which then reveals itself to be much too difficult.

“Great. Now go ahead and add 6 + 17.”

“Great. Now go ahead and find the slope between (-5, 3) and (5, 10,003).”

“Great. I’m going to show you the number 5,203,584,109,402,580 for ten seconds. Remember it as accurately as you can.”

These tasks seem easy given our current toolset but are actually quite hard, which can lead to curiosity about stronger counting strategies, a generalized slope formula, and scientific notation, respectively.

Create an open middle.

When you look at successful, engaging video games (even the fake-world games with no real-world application) they generally start with the same initial state and the same goal state, but how you get from one to the other is left to you. This gives the student the sense that her path is self-determined, rather than pre-determined, that she’s autonomous. (See: Deci; Csikszentmihalyi.)

Sudoku. You start with a partially-completed game board and your goal is to complete it. You can wander down some dead ends as you accomplish the task. How you get there is up to you.

Jinx Puzzle. You get to choose your number. It will be different from other people’s numbers. What you start with is up to you.

Area v. Perimeter. You get to choose your rectangle. It will be different from other people’s rectangles. What you start with is up to you.

I’m not recommending “open problems” here because the language there is too flexible to be meaningful and too accommodating of a lot of debilitating student frustration. I’m not recommending you throw a video on the wall and let students take it wherever they want. I’m recommending that you’re exceptionally clear about where your students are and where they’re going but that you leave some of the important trip-planning to them.

Give students exactly the right kind of feedback in the right amount at the right time.

Easy, right? Feedback has been well-studied from the perspective of student learning but feedback’s effect on student interest is complicated. Some of you have recommended “immediate” feedback in the comments, but this may have the effect of prodding students down an electrified corridor where every deviation from a pre-determined path will register an alarm, creating a very closed middle. Students need to know if they’re on the right track while simultaneously preserving their ability to go momentarily off on the wrong track. This isn’t simple, but the best games and the best tasks maintain that balance.

Four Fours. You arrange the fours into whatever configuration you want and then you check your answer. The feedback isn’t immediate. But you can check it yourself.

Area v. Perimeter. You develop a theory about the impossible points then later test that theory out on different rectangles and coordinates.

Sudoku. You don’t receive feedback immediately after you write a number in a box. But eventually you’re able to decide if the gameboard you’ve created matches the rules of the game.

Conclusion

You could very well say that these rules apply to “real world” tasks just as well as they apply to the world of pure mathematics. Exactly right! In the first post of this series, I said that “the real world-ness of [an engaging real-world] task is often its least essential element.” Real-world tasks are sometimes the best way to accomplish the pedagogy I’ve summarized here but it’s a mistake to assume that the “real world,” itself, is a pedagogy.

As I’ve tried to illustrate in this post with different existence proofs, it’s also a mistake to assume that pure math is hostile to student curiosity. The recommendations from these researchers can all be accomplished as easily with numbers and symbols and shapes as with two trains leaving Chicago traveling in opposite directions.

I wonder, though, if declaring some content “boring bits” gives up the game. (If they are *truly* and irredeemably boring, why teach them?) First, we need to understand these “bits” better and think about big-picture coherence.

Consider a jigsaw-puzzle. A missing piece is “interesting” quite apart from its shape or color (although they may be independent sources of interest). What matters is that it fits! Similarly, no one salivates over dates and locations in history. Yet we care about these facts because they allow us to tell stories, discover relationships, and find patterns. It is silly to complain that a particular fact is “boring” on its own. Here again, what matters it how it fits.

I like that these six methods have nothing to do with making math easy. This is often the identified goal that students and teachers (and countless online videos) work towards, particularly in the context of students that have failed math in the past. Having success is enough of a motivator for students to push through some material that is not particularly engaging, it does not have staying power.

Jason Dyer challenges the group to apply these principles to fifth-degree polynomial inequalities:

I don’t suppose we could take something extra-”boring” and try to do a makeover? Sort of like Dan’s makeover series except start from the hardest point possible?

I am having to currently teach working out polynomial inequalities like 2x^5 + 18x^4 + 40x^3 < 0. There's an intense amount of drudge and fiddly bits and the extra pain of having a lot of steps to do. Any ideas?

Hi there and happy 2014. I’m currently flying from sunny Mountain View, CA, into the heart of the polar vortex for some work with math educators in Madison, WI. That’ll be the first of 50-ish workshops and talks I’ll be offering over 2014. You can find the rest of that list on my presentation page. I’ve posted links to sessions that have open registrations. (Some later dates haven’t opened registration yet.) If you have any questions, feel free to drop a comment here or an e-mail at dan@mrmeyer.com. Do say hi if we’re in the same area and tell me something interesting you’ve learned about teaching this year.